1. Anna Philippou Department of Computer Science University of Cyprus Joint work with Mauricio Toro Department of Comp. Sc. EAFIT University Christina Kassara Department of Biology University of Patras Spyros Sfenthourakis Department of Biology University of Cyprus Synchronous Parallel Composition in Process Calculus for Ecological Models

2. Motivation and Background Population Ecology – Population dynamics – Conservation schemes, species reintroduction Individual-based modeling vs. population-based modeling Formal methods for ecological systems – Process calculi, P-systems, cellular automata 2/25

3. Motivation and Background Previous work – PALPS (Process Algebra with Locations for Population Systems) a discrete-time, probabilistic process calculus with locations – Translation from PALPS to model checker PRISM – Application of the framework for studying population dynamics through model checking Limitations – State space explosion – Interleaving semantics in process calculi vs. phase-based execution considered by ecologists 3/25

4. Illustration by example 4/25 … move rep

5. Illustration by example 5/25 = move = rep …

6. Illustration by example = move = rep … 6/25

7. Illustration by example Intention: individuals execute their dispersal and reproduction phases simultaneously. Proposed solution – Syntax Group individuals into populations – Semantics Synchronous parallel composition E.g. – System with k individuals: S  k  – state space for k individuals 7/25 move k rep k

8. S-PALPS An extension of PALPS: a discrete-time, probabilistic process algebra with locations World consisting of a set of locations each featuring its own environmental characteristics and inhabited by populations. World implemented as a graph: 8/25

9. S-PALPS Basic entities – Channels: a, b, … can be used in input position, a, b, or in output position a, b – Locations: l 1, l 2, … And a set of logical expressions, e, referring to properties of locations – Species: s 1, s 2, … Individuals – Modelled as processes – Possess a species and a location – Locations may change dynamically – Engage in activities (e.g. reproduction, dispersal, predation, death) 9/25

10. Syntax (1) 10/25

11. Syntax (2) 11/25

12. S-PALPS Example 12/25

13. S-PALPS Semantics 13/25

14. Selected Rules – Population Level (ACT) (MOVE) 14/25

15. Selected Rules – System Level (TICK) (SYNCH) 15/25

16. Model checking S-PALPS models S-PALPS  models where probabilistic and nondeterministic behavior co-exists PRISM  model checker for probabilistic systems (Markov Decision Processes, Discrete Markov Chains, Continuous MCs) Prism language – State-based – Guarded commands – Set of modules that communicate with each other on shared actions (CSP-style communication) 16/25

17. Encoding PALPS into PRISM To translate SPALPS into the PRISM language – Each state of a located population is a module – A module has a variable that counts the number of individuals at the specific state – the execution flow is facilitated by a local variable – all modules synchronize via one of the following actions: synch – executed when a nondeterministic transition takes place tick – executed when a timed action takes place prob – executed when a probabilistic transition takes place 17/25

18. global s p :[0,max] init 5 global s Q :[0,max] init 0 module P st:[0..5] init 1; n P :[0..max] [] (st=1)&(s P >0)  (tact’=0)& (st’=2)&(n p ’=s P ); [] (st=1)&(s P =0)  (st’=2); [synch] (st=2)  (st’=3); [](st=3)&(pact=0)  (s P ’=s p – n p ) &(s Q ’=s Q +n P ) [prob] (st=3)&(pact=1)  … module Q … Encoding S-PALPS into PRISM – A Snapshot 18/25

19. Correctness of the encoding 19/25

20. Case Study – The Eleonora Falcon Migrant species that breeds in the Mediterranean sea Local conservation importance Three types of individuals – Juveniles – Young adults (sexual maturity at the age of 4) – Mature adults (more likely to choose a protected nest) Parameters – Probability of an individual to return to the island to breed (mortality) – Availability of less-exposed nests – Probabilities of offspring survival in exposed/less-exposed nests 20/25

21. The life cycle 21/25

22. The model in S-PALPS 22/25

23. Analysis in PRISM Verification of probabilistic temporal PCTL properties, e.g. – Probability of extinction of the population in the next 10 years is less than a certain threshold p e – The total average number of individuals of species s at time unit t Semantics of PRISM model checking – Defined over Markov Decision Processes: Computes minimum and maximum probabilities – Approximation defined over Discrete-Time Markov Chains: Computes reward-based properties, steady state and reachability Simulation – Explore random paths of execution – Search for deadlocks using Prism simulation – Perform model-checking by simulation (statistical model checking) 23/25

24. Representative results Impact of changing the offspring survival rates on the size of the population Results indicate a fair degree of stability in the evolution of the species and a relative insensitivity to small changes in local conditions 24/25

25. Conclusions and future work S-PALPS – Synchronous parallel composition – more faithful models – Reduction of the state space of systems – Case study – promising results Current/future work: – Translation from S-PALPS to PRISM has been automated – Improve the translation: generality vs. efficiency – Further calibrate our model and take into account more aspects of interest to biologists – Apply to other case studies – Mean field semantics à la CCS within a spatially-explicit framework 25/25